Cause of Brownian motion's indeterminacy

No, there is no simple connection.

In quantum mechanics, the position of the particle is represented by a wave function so that the particle is "everywhere" and only after measurement can we localise it. This localisation has an error which is not due to the instrument's precision but rather is intrinsic in the "shape" of the wave function and indeed can be predicted regardless of the measurement apparatus.

In the case of Brownian motion, we are talking about a big particle moving erratically. However, its position and momentum are well-defined, the reason Brownian motion is modelled as a stochastic process is that it is too hard to measure it with enough precision. But if you had a very fast microscope, you could measure its position in space and time with increasing precision.

There is only one last point: because everything is at some level described by quantum mechanics, there is a sort of "leftover" indeterminacy in everything, including Brownian motion. However, for objects of bigger scale (e.g. a colloidal particle) that can often be completely neglected. Predicting or measuring exactly what fluid molecule is pushing the particle might indeed be harder and quantum-like, but measuring the position and momentum of the Brownian particle, it being a classical particle, is just a resolution issue.